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Parabolic Monge-Ampère equations on Riemannian manifolds. (English) Zbl 0895.58053
The author considers on a compact Riemannian manifold $$(M,g)$$ the parabolic Monge-Ampère equation ${{\partial}\over{\partial t}}\varphi (x,t)=\log\left({{\det\Bigl(g(x)+\text{ Hess}\varphi (x,t)\Bigr)}\over{\det g(x)}}\right)-\lambda\varphi (x,t)-f(x),\quad \varphi(x,0)=\varphi_0(x),$ where $$\lambda\in\mathbb{R}$$ is a parameter and $$\varphi_0,f\in C^\infty(M,\mathbb{R}).$$ She shows global existence in time, independently of $$\lambda.$$ Moreover, when $$\lambda >0$$, she proves that $$\varphi_t=\varphi(\cdot,t)$$ exponentially converges as $$t\to +\infty$$ to a solution $$\varphi_\infty$$ of the stationary problem and that, when in addition $$f=0$$, one has the existence of $$\delta, c>0$$ (depending on $$\varphi_0$$ and $$| \nabla^i\varphi| _\infty$$, $$i=0,1,2,3)$$ such that $\int_M\bigl(\varphi_t -\overline{\varphi_t}\bigr)^2 d\text{ Vol}_g\leq c\exp\Bigl(-2(\mu_1+\lambda+e^{-\delta t})t\Bigr),$ where $$\overline\varphi$$ denotes the mean value of $$\varphi$$ and $$\mu_1$$ is the first eigenvalue of the Laplacian.

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
##### Keywords:
parabolic Monge-Ampère equations
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##### References:
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